Metamath Proof Explorer

Theorem qexpclz

Description: Closure of integer exponentiation of rational numbers. (Contributed by Mario Carneiro, 9-Sep-2014)

Ref Expression
Assertion qexpclz 
|- ( ( A e. QQ /\ A =/= 0 /\ N e. ZZ ) -> ( A ^ N ) e. QQ )

Proof

Step Hyp Ref Expression
1 qsscn 
 |-  QQ C_ CC
2 qmulcl 
 |-  ( ( x e. QQ /\ y e. QQ ) -> ( x x. y ) e. QQ )
3 1z 
 |-  1 e. ZZ
4 zq 
 |-  ( 1 e. ZZ -> 1 e. QQ )
5 3 4 ax-mp 
 |-  1 e. QQ
6 qreccl 
 |-  ( ( x e. QQ /\ x =/= 0 ) -> ( 1 / x ) e. QQ )
7 1 2 5 6 expcl2lem 
 |-  ( ( A e. QQ /\ A =/= 0 /\ N e. ZZ ) -> ( A ^ N ) e. QQ )